Multiple-input multiple-output (MIMO) technology exploits the spatial components of the wireless channel to provide capacity gain and increased link robustness. After almost a decade of research, multiple-input multiple-output (MIMO) has finally been adopted in several standards including IEEE 802.16e-2005 and IEEE 802.11n; products based on draft standards are already shipping. MIMO is often combined with OFDM (orthogonal frequency division multiplexing), a type of digital modulation that makes it easy to equalize broadband channels.
In MIMO communication systems, at the transmitter, data are modulated, encoded, and mapped onto spatial signals, which are transmitted from the multiple transmit antennas. A main difference with non-MIMO communication systems is that there are many different spatial formatting modes for example beamforming, preceding, spatial multiplexing, space-time coding, and limited feedback preceding, among others (see A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications, 40 West 20th Street, New York, N.Y., USA: Cambridge University Press, 2003). The spatial formatting techniques have different performance (in terms of capacity, goodput, achievable rate, or bit error rate for example) in different channel environments. Consequently, there has been interest in adapting the spatial transmission mode based on information obtained about the channel.
One especially effective technique is known as closed-loop MIMO communication, where channel state information or other channel-dependent information is provided from the receiver to the transmitter through a feedback link. This information is used to customize the transmitted signal to the current propagation conditions to improve capacity, increase diversity, reduce the deleterious effects of fading, or support more users in the communication link for example. Because the bandwidth of the feedback link is low, techniques for quantizing channel state information and other receiver information have become increasingly important. This research area dealing with quantizing channel state information and other channel-dependent parameters is broadly known as limited feedback communication (see D. J. Love, R. W. Heath, Jr., W. Santipach, and M. L. Honig, “What is the Value of Limited Feedback for MIMO Channels?” IEEE Communications Magazine, vol. 42, no. 10, pp. 54-59, October 2003). The concept of limited feedback can be applied to any communication system but it is especially valuable in MIMO communication systems.
Limited feedback precoding is a preferred embodiment of the limited feedback concept for MIMO communication channels. The main concept of limited feedback precoding is that an index of a quantized precoding matrix from a predetermined codebook of codewords in the form of precoding vectors or matrices (known at both the transmitter and receiver), is determined at the receiver and sent back to the transmitter over the feedback link. The determination of the preferred index can be made based on several different design criteria including maximum capacity, maximum goodput, minimum error rate, or minimum distortion for example. While the optimum index can be computed based on any number of measurements made at the receiver including the channel state estimates and statistics of the channel state like the mean and covariance, computations based directly on the channel state estimates are known to have the best performance.
The codebook employed in a limited feedback technique is known to have an impact on the eventual system performance. Larger codebooks, which require more bits to represent the index, are generally of higher resolution and have better performance at the expense of requiring more feedback to send back the codebook index. Smaller codebooks require fewer bits to represent the index of the chosen codeword, thus entailing reduced feedback overhead at the expense of lower resolution. Furthermore, larger codebooks require more storage space, which may tax the storage capabilities of the transmitter and the receiver. Because the feedback channel constitutes system overhead, there is a tension between using more feedback overhead to obtain higher resolution and using less feedback to reduce the penalty due to feedback overhead.
Many different codebook designs have been proposed in the literature for use in limited feedback precoding systems. A prominent example are Grassmannian codebooks (see D. Love and R. W. Heath Jr., “Limited feedback unitary precoding for orthogonal space-time block codes,” IEEE Trans. Signal Processing, vol. 53, no. 1, pp. 64-73, 2005; D. Love, J. Heath, R. W., and T. Strohmer, “Grassmannian beamforming for multiple-input multiple-output wireless systems,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2735-2747, 2003; and D. Love and J. Heath, R. W., “Limited feedback unitary precoding for spatial multiplexing systems,” IEEE Trans. Inform. Theory, vol. 51, no. 8, pp. 2967-2976, 2005). With Grassmannian codebooks, the codebook is designed to correspond to a good packing on the Grassmann manifold, essentially maximizing the minimum subspace distance measured using for example the Chordal distance, Fubini-Study distance, and projection 2-norm distance. These codebooks are optimal in some sense but only exist in special cases and are extremely difficult to compute even when they exist.
Vector quantization concepts have also been used to design codebooks (see A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, “Efficient use of side information in multiple-antenna data transmission over fading channels,” IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1423-1436, October 1998; and J. C. Roh and B. D. Rao, “Transmit beamforming in multiple-antenna systems with finite rate feedback: a VQ-based approach,” IEEE Trans. Inform. Theory, vol. 52, no. 3, pp. 1101-1112, March 2006). The idea is that the codebook is constructed using an iterative technique according to a distortion measure like subspace distance, average capacity, or bit error rate for example. This approach can be used to design a codebook of any size but the codebook usually lacks structure to allow efficient storage. Further such codebooks may not be globally optimal since an iterative algorithm is employed.
Other codebooks have been proposed based on the Fourier transform (see for example D. Love and R. W. Heath Jr., “Limited feedback unitary precoding for orthogonal space-time block codes,” IEEE Trans. Signal Processing, vol. 53, no. 1, pp. 64-73, 2005; and R1-072235, Samsung, “Codebook design for 4tx SU MIMO,” 3GPP TSG RAN WG1 49, Kobe, Japan, 7-11 May, 2007. Available at http://www.3gpp.org/ftp/tsg ran/WG1 RL1/TSGR1 49/Docs/R1-072235.zip). These codebooks can be stored efficiently and have properties that may simplify computation. They only exist though for small codebook sizes.
Yet other codebooks have been designed based on Kerdock codes or mutually unbiased bases (T. Inoue and R. W. Heath Jr., “Kerdock codes for limited feedback MIMO systems,” Mar. 30-Apr. 4, 2008, Proc. of the IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Las Vegas, Nev.). This codebook is constructed from multiple sets of unitary matrices such that the maximum minimum inner product between columns is maximized. They have good properties that make them easy to search and a quarternary alphabet that makes them easy to store but the codebook size is limited.
Other codebooks have been suggested that have a nested structure allowing them to work with a different number of substreams including one, two, three, and four streams. An example of this is the Householder codebook design (R1-072201, “Way forward on 4-tx antenna codebook for su-mimo,” 3GPP TSG RAN WG1 49, Kobe, Japan, 7-11 May, 2007. Available at http://www.3gpp.org/ftp/tsg ran/WG1 RL1/TSGR1 49/Docs/R1-072201.zip) where a beamforming codebook like a Grassmannian codebook is to compute Householder reflection matrices that are used to construct more complex codebooks using columns from these matrices. These codebooks have some advantages that they can be easy to store since the coefficients of the codewords can be represented with low precision, though the codebooks are somewhat small.
Other codebook designs have been suggested to exploit adaptive feedback to reduce the amount of feedback. One approach is to take advantage of spatial and temporal correlation to reduce the amount of feedback required. For example one approach is to use a set of possible codebooks. The best codebook changes over time and is adaptively sent back to the transmitter (see for example B. Mondal and R. W. Heath, Jr., “Channel Adaptive Quantization for Limited Feedback MIMO Beamforming systems,” IEEE Trans. on Signal Processing, vol. 54, no. 12., pp. 4741-4740, December 2006). This approach requires additional overhead to signal the switch between codebooks in addition to the extra storage space to store the set of possible codebooks. In another approach to adaptive feedback (B. C. Banister and J. R. Zeidler, “Feedback Assisted Stochastic Gradient Adaptation of Multiantenna Transmission,” IEEE Transactions on Wireless, vol. 4, no. 3, pp. 1121-1135, May 2005), gradients in an adaptive algorithm are quantized and sent back to the transmitter. This algorithm may take a long time to converge.
Another approach for codebooks that facilitate adaptation uses a localized codebook is combined with a non-local codebook to facilitate adaptation to spatial correlation (R. Samanta and R. W. Heath Jr., “Codebook Adaptation for Quantized MIMO Beamforming Systems,” Proc. of the Asilomar Conference, October 2005, pp. pp. 376-380). This paper describes a localized codebook, scaling and rotation operations, and an adaptation mechanism. In this prior work localized codebooks are described but are used only when the source distribution is determined to be suitably localized. Thus there is an algorithm that effectively chooses the base codebook and the optimum radius for the localized codebook, then that codebook used during several quantization periods. Design criteria for finding the localized codebooks were not discussed. Their application to multiuser communication was not mentioned. Similarly, this concept was used in reference V. Raghavan, R. W. Heath, and A. M. Sayeed, “Systematic Codebook Designs for Quantized Beamforming in Correlated MIMO Channels,” IEEE J. Select. Areas Commun., vol. 25, no. 7, pp. 1298-1310, September 2007. The key concept in this prior work is to show when correlated channels have sufficiently localized eigenvectors that can be reasonably quantized with a localized codebook. Design criteria for finding the localized codebooks were not discussed. Their application to multiuser communication was not mentioned.
Yet other codebooks are designed so that they can be searched efficiently (see for example D. J. Ryan, I. V. L. Clarkson, I. B. Collins, D. Guo, and M. L. Honig, “QAM Codebooks for Low-Complexity Limited Feedback MIMO Beamforming,” Proc. of ICC 2007, pp. 4162-4167). In this case special mathematical structure in the codebook permits algorithms that can perform the quantization efficiently, at the expense of a larger codebook size for the same performance with other codebook techniques. The ability to search the codebook is especially important for high resolution limited feedback, which requires large codebook sizes, because without special structure a brute-force search of the codebook is required.
High resolution, or larger, codebooks are especially important for a type of MIMO communication known as multiuser MIMO or MU-MIMO (see D. Gesbert, M. Kountouris, R. W. Heath, Jr., C. B. Chae, and T. Salzer, “From Single user to Multiuser Communications: Shifting the MIMO paradigm,” IEEE Signal Processing Magazine, Vol. 24, No. 5, pp. 36-46, October, 2007 and the references therein). In MU-MIMO, multiple users share the propagation channel. In what is known as the downlink or broadcast MU-MIMO channel, information is sent to multiple users via specially designed transmit beamformers or precoders determined based on channel state information. Users send information about their channel state through an uplink feedback channel.
The concept of limited feedback has been used in MU-MIMO communication systems (see for example N. Jindal, MIMO Broadcast Channels with Finite Rate Feedback, IEEE Trans. Information Theory, Vol. 52, No. 11, pp. 5045-5059, November 2006) to compress quantized information sent on the uplink. A main conclusion of this paper is that the codebook size in a MU-MIMO communication system measured in terms of number of bits grows in proportion to the number of users in the system. So if a single user MIMO system requires a 6 bit codebook, a MU-MIMO system supporting transmission to two users might require at least a 12 bit codebook (64 codewords versus 4,096 codewords). The codebook size also grows as a function of the operating SNR due to an error floor effect. Scheduling the best users can reduce this effect (see for example T. Yoo, N. Jindal, and A. Goldsmith, Multi-Antenna Downlink Channels with Limited Feedback and User Selection, IEEE Journal Sel. Areas in Communications, Vol. 25, No. 7, pp. 1478-1491, September 2007). Nonetheless, MU-MIMO requires high resolution limited feedback codebooks. Codebook designs have not been extensively investigated for MU-MIMO communication systems. The codebook designs discussed already are typically small, without the required resolution for MU-MIMO, or are large but require high complexity to search.
Practical codebook design require that several criteria are met, which is not solved. They should have low storage requirements. This means that high precision is not required to store each codebook entry. Unfortunately, much of the prior work (Grassmannian and vector quantization codebooks for example) does not satisfy this criterion.
Efficient algorithms with reasonable computational complexity should be available to efficiently search for an optimum codeword in the codebook. Unfortunately, most prior work (with the exception of the work by D. J. Ryan et. al.) gives codebooks that do not facilitate especially efficient codeword search. Most existing approaches for limited feedback codebook design are single shot in that they quantize the current channel state without considering the previous channel state. Adaptive codebook strategies, though, could be used to improve performance by only compressing changes in the channel state. Special codebooks are needed to allow efficient adaptive feedback but have only seen limited development (the work by Samanta et. al. for example).
Finally, it would be advantageous if codebooks could be applied to both single user and multiuser communication settings. Unfortunately, single user codebooks are usually designed to be smaller and are not big enough to support the resolution required by multiuser codebooks.